Utility-Driven Cooperation and Emergent Group Diversity in the N-Player Stag Hunt Game: A Framework Relevant to Collective Biological Behavior.

The coordination dilemma inherent in the N-player Stag Hunt game provides a powerful framework for studying the emergence of cooperation in social and biological systems. Traditional analyses of this game emphasize the tension between individual risk aversion and collective benefit. In this study, we extend the classical N-player Stag Hunt model by incorporating utility functions that formalize how intra-group cooperation and inter-group competition interact. Using this framework, we derive the Nash equilibria analytically, revealing how different utility formulations can generate a wide spectrum of equilibrium outcomes, ranging from fully noncooperative behavior to the formation of evenly or unevenly sized cooperative groups. Our findings show that this diversity is not merely a consequence of stochasticity or network topology but can systematically emerge from underlying variations in utility functions. Beyond social decision-making, these results provide a general mathematical perspective on biological collective behaviors in which group size and stability play a functional role, such as microbial aggregation, multicellular coordination, and the observed variability in cluster sizes of circulating tumor cells. By highlighting how utility-driven preferences shape the emergence of group structures, the model offers a potential framework for interpreting size-dependent cooperation in diverse adaptive systems.